\section{Proof of Spectral Stability and Convergence Theorem}
\label{app:spectral-proof}

We provide a complete proof of \Cref{thm:spectral-stability} establishing existence, uniqueness, and exponential convergence to the fixed point of the hierarchical cooperation dynamics. The proof employs Perron-Frobenius theory for non-negative matrices, spectral decomposition of the coupling matrix, and contraction mapping arguments.

\subsection{Preliminaries and Notation}

Consider the discrete-time dynamical system governing order parameter evolution:
\begin{equation}
    \Phi(t+1) = M\Phi(t) + b,
    \label{eq:linear-dynamics-app}
\end{equation}
where $\Phi(t) \in \mathbb{R}^L$ is the vector of order parameters at hierarchical levels $\ell = 1,\ldots,L$, $M \in \mathbb{R}^{L \times L}$ is the coupling matrix defined in \eqref{eq:coupling-matrix}, and $b \in \mathbb{R}^L$ represents external inputs and baseline activity.

The coupling matrix has the explicit form
\begin{equation}
    M_{\ell\ell'} = \beta_\ell J_{\text{eff}}^{(\ell)} \delta_{\ell\ell'} + \beta_\ell K_{\text{eff}}^{(\ell,\ell')} (1-\delta_{\ell\ell'}),
\end{equation}
where $J_{\text{eff}}^{(\ell)} \geq 0$ represents effective intra-level coupling strength, $K_{\text{eff}}^{(\ell,\ell')} \geq 0$ captures inter-level coordination strength, and $\beta_\ell > 0$ are inverse temperature parameters.

\subsection{Perron-Frobenius Theory Background}

We first recall essential results from Perron-Frobenius theory that underpin our convergence analysis.

\begin{theorem}[Perron-Frobenius for Non-negative Matrices]
\label{thm:perron-frobenius}
Let $A \in \mathbb{R}^{n \times n}$ be a non-negative matrix ($A_{ij} \geq 0$). Then:
\begin{enumerate}
    \item The spectral radius $\rho(A) = \max_k |\lambda_k(A)|$ is an eigenvalue of $A$.
    \item There exists a non-negative eigenvector $v \geq 0$ such that $Av = \rho(A)v$.
    \item If $A$ is irreducible, then:
    \begin{enumerate}
        \item $\rho(A)$ is a simple eigenvalue (algebraic and geometric multiplicity one)
        \item The corresponding eigenvector $v$ is strictly positive: $v > 0$
        \item $\rho(A)$ is the only eigenvalue with a non-negative eigenvector
    \end{enumerate}
    \item If $A$ is irreducible and aperiodic, then all eigenvalues $\lambda_k$ satisfy $|\lambda_k| < \rho(A)$ except $\lambda_0 = \rho(A)$.
\end{enumerate}
\end{theorem}

A matrix $A$ is \emph{irreducible} if its associated directed graph is strongly connected. A matrix is \emph{aperiodic} if $A_{ii} > 0$ for some $i$ (existence of self-loops).

\subsection{Verification of Regularity Conditions}

We verify that the coupling matrix $M$ satisfies the conditions required for Perron-Frobenius theory.

\textbf{Non-negativity.} By construction, $J_{\text{eff}}^{(\ell)} \geq 0$, $K_{\text{eff}}^{(\ell,\ell')} \geq 0$, and $\beta_\ell > 0$, ensuring $M_{\ell\ell'} \geq 0$ for all $\ell, \ell'$.

\textbf{Irreducibility.} The hierarchical coupling graph is strongly connected by assumption: information flows both upward through aggregation operators $A_\ell$ and downward through actuation operators $D_\ell$. This bidirectional exchange creates paths between all levels.

\textbf{Aperiodicity.} Self-transitions exist at each level due to persistence of local state and intra-level coupling: $M_{\ell\ell} = \beta_\ell J_{\text{eff}}^{(\ell)} > 0$ for all $\ell$. This guarantees aperiodicity.

\subsection{Fixed Point Existence and Uniqueness}

\begin{proposition}[Fixed Point Properties]
\label{prop:fixed-point-app}
If $\rho(M) < 1$, then:
\begin{enumerate}
    \item The matrix $(I - M)$ is invertible
    \item The system \eqref{eq:linear-dynamics-app} admits a unique fixed point $\Phi^* = (I - M)^{-1}b$
    \item The fixed point satisfies $\Phi^* \geq 0$ when $b \geq 0$
\end{enumerate}
\end{proposition}

\begin{proof}
(1) Suppose $(I - M)v = 0$ for some $v \neq 0$. Then $Mv = v$, meaning $1$ is an eigenvalue of $M$. But $\rho(M) < 1$ implies all eigenvalues satisfy $|\lambda_k| < 1$, contradicting $\lambda = 1$. Therefore $(I - M)$ is invertible.

(2) The fixed point equation $\Phi^* = M\Phi^* + b$ is equivalent to $(I - M)\Phi^* = b$. By invertibility, $\Phi^* = (I - M)^{-1}b$ exists and is unique.

(3) When $\rho(M) < 1$, the Neumann series converges:
\begin{equation}
    (I - M)^{-1} = \sum_{k=0}^{\infty} M^k.
\end{equation}
Since $M \geq 0$, all powers $M^k \geq 0$, implying $(I - M)^{-1} \geq 0$. For $b \geq 0$, we obtain $\Phi^* = (I - M)^{-1}b \geq 0$.
\end{proof}

\subsection{Spectral Decomposition and Eigenvalue Structure}

The spectral decomposition of $M$ provides the foundation for analyzing convergence dynamics. Since $M$ is irreducible and aperiodic with $\rho(M) < 1$, Theorem \ref{thm:perron-frobenius} guarantees:
\begin{itemize}
    \item Dominant eigenvalue $\lambda_0 = \rho(M)$ with multiplicity one
    \item Strictly positive right eigenvector $v_0 > 0$: $Mv_0 = \rho(M)v_0$
    \item Positive left eigenvector $u_0 > 0$: $u_0^T M = \rho(M)u_0^T$
    \item Spectral gap $\gamma = \rho(M) - \max_{k \geq 1} |\lambda_k| > 0$
\end{itemize}

We normalize eigenvectors such that $u_0^T v_0 = 1$ and construct the spectral decomposition:
\begin{equation}
    M = \rho(M) v_0 u_0^T + \sum_{k=1}^{L-1} \lambda_k v_k u_k^T,
    \label{eq:spectral-decomp-app}
\end{equation}
where $\{v_k, u_k\}_{k=1}^{L-1}$ form bi-orthogonal bases for the remaining eigenspaces with $|\lambda_k| < \rho(M)$.

\subsection{Exponential Convergence Rate}

\begin{theorem}[Exponential Convergence]
\label{thm:exponential-convergence-app}
Under the conditions of \Cref{thm:spectral-stability}, the deviation from the fixed point satisfies:
\begin{equation}
    \|\Phi(t) - \Phi^*\| \leq C \rho(M)^t \|\Phi(0) - \Phi^*\|,
\end{equation}
where $C = \kappa(V)$ is the condition number of the eigenvector matrix $V = [v_0, v_1, \ldots, v_{L-1}]$.
\end{theorem}

\begin{proof}
Define the deviation $\delta(t) = \Phi(t) - \Phi^*$. Since $\Phi^*$ is a fixed point:
\begin{equation}
    \delta(t+1) = \Phi(t+1) - \Phi^* = M\Phi(t) + b - (M\Phi^* + b) = M\delta(t).
\end{equation}
Iterating yields $\delta(t) = M^t \delta(0)$.

Using the spectral decomposition \eqref{eq:spectral-decomp-app}:
\begin{equation}
    M^t = \rho(M)^t v_0 u_0^T + \sum_{k=1}^{L-1} \lambda_k^t v_k u_k^T.
\end{equation}

Express the initial deviation in the eigenbasis: $\delta(0) = \sum_{k=0}^{L-1} c_k v_k$ where $c_k = u_k^T \delta(0)$. Then:
\begin{equation}
    \delta(t) = M^t \delta(0) = \rho(M)^t c_0 v_0 + \sum_{k=1}^{L-1} \lambda_k^t c_k v_k.
\end{equation}

For transient dynamics starting from arbitrary $\Phi(0)$:
\begin{equation}
    \|\delta(t)\| \leq \sum_{k=0}^{L-1} |\lambda_k|^t |c_k| \|v_k\| \leq \rho(M)^t \sum_{k=0}^{L-1} |c_k| \|v_k\|.
\end{equation}

Using $\sum_{k} |c_k| \|v_k\| \leq \kappa(V) \|\delta(0)\|$ where $\kappa(V) = \|V\| \|V^{-1}\|$:
\begin{equation}
    \|\delta(t)\| \leq \rho(M)^t \kappa(V) \|\delta(0)\| = C \rho(M)^t \|\delta(0)\|.
\end{equation}
\end{proof}

\subsection{Mixing Time Bound}

The mixing time $t_{\text{mix}}(\epsilon)$ quantifies how long the system requires to approach the fixed point within tolerance $\epsilon$.

\begin{definition}[Mixing Time]
The $\epsilon$-mixing time is:
\begin{equation}
    t_{\text{mix}}(\epsilon) = \min\{t \geq 0 : \|\Phi(t') - \Phi^*\| \leq \epsilon \|\Phi(0) - \Phi^*\| \text{ for all } t' \geq t\}.
\end{equation}
\end{definition}

\begin{theorem}[Mixing Time Bound]
\label{thm:mixing-time-app}
Under the conditions of \Cref{thm:spectral-stability}, the mixing time satisfies:
\begin{equation}
    t_{\text{mix}}(\epsilon) \leq \frac{\ln(C/\epsilon)}{\gamma},
\end{equation}
where $\gamma = 1 - \rho(M)$ is the spectral gap.
\end{theorem}

\begin{proof}
From Theorem \ref{thm:exponential-convergence-app}, we require $C \rho(M)^t \|\delta(0)\| \leq \epsilon \|\delta(0)\|$, which simplifies to $\rho(M)^t \leq \epsilon/C$. This holds when:
\begin{equation}
    t \geq \frac{\ln(C/\epsilon)}{-\ln \rho(M)}.
\end{equation}

For $\rho(M) < 1$, we use $-\ln \rho(M) = -\ln(1 - \gamma) \geq \gamma$ where $\gamma = 1 - \rho(M)$, giving the stated result.
\end{proof}

\textbf{Interpretation.} The spectral gap $\gamma = 1 - \rho(M)$ controls convergence speed: larger gaps produce faster mixing, while small gaps (near-critical regime, $\rho(M) \to 1^-$) lead to critical slowing down with diverging mixing times $t_{\text{mix}} \sim \gamma^{-1}$.

\subsection{Discussion of Regularity Conditions}

\subsubsection{Non-negativity: $M_{\ell\ell'} \geq 0$}

Non-negativity reflects the cooperative nature of hierarchical coordination: increased activity at one level stimulates activity at connected levels.

\textbf{Violation consequences.} Negative couplings introduce competitive interactions. Oscillatory instabilities may emerge from complex eigenvalues.

\subsubsection{Irreducibility: Strongly Connected Hierarchy}

Irreducibility ensures all levels participate in the global dynamics—no isolated subsystems exist.

\textbf{Violation consequences.} Reducible matrices decompose into blocks evolving independently. Information silos emerge, preventing coordinated global behavior.

\subsubsection{Aperiodicity: Self-Transitions Exist}

Aperiodicity eliminates cyclic behavior where the system rotates through states without settling.

\textbf{Violation consequences.} Periodic matrices have eigenvalues on the unit circle. The system oscillates without converging to a fixed point.

\subsubsection{Spectral Stability: $\rho(M) < 1$}

The spectral radius condition is the fundamental stability requirement ensuring the iterative map is a contraction.

\textbf{Physical interpretation.} The condition $\rho(M) < 1$ means cumulative feedback strength across hierarchical levels remains subcritical. At $\rho(M) = 1$, the system undergoes a phase transition; for $\rho(M) > 1$, positive feedback dominates, leading to divergence.

\textbf{Critical behavior near $\rho(M) \to 1^-$.} As spectral stability weakens, $\gamma = 1 - \rho(M) \to 0$, producing:
\begin{itemize}
    \item Diverging mixing time: $t_{\text{mix}} \sim \gamma^{-1}$
    \item Amplified fluctuations: $\text{Var}[\Phi_\ell] \sim \gamma^{-1}$
    \item Enhanced susceptibility: $\chi \sim \gamma^{-1}$
\end{itemize}

\subsection{Summary}

We have established complete proof of Theorem \ref{thm:spectral-stability} through:
\begin{enumerate}
    \item Verification that $M$ satisfies Perron-Frobenius conditions (non-negativity, irreducibility, aperiodicity)
    \item Proof of unique fixed point existence via invertibility of $(I - M)$ when $\rho(M) < 1$
    \item Spectral decomposition revealing eigenvalue structure with spectral gap $\gamma = 1 - \rho(M)$
    \item Derivation of exponential convergence $\|\Phi(t) - \Phi^*\| \leq C\rho(M)^t\|\delta(0)\|$ via iterated dynamics
    \item Mixing time estimate $t_{\text{mix}}(\epsilon) \leq \ln(C/\epsilon)/\gamma$ from convergence threshold
    \item Discussion of regularity conditions and consequences of violations
\end{enumerate}

The proof unifies fixed-point theory, spectral analysis, and dynamical systems, providing both existence guarantees and quantitative convergence rates essential for designing stable hierarchical cooperation systems.
